Example
Example 78.1. Correlations among Physical Variables
The following data are correlations among eight physical variables as given by Harman (1976). The first PROC VARCLUS run clusters on the basis of principal components, the second run clusters on the basis of centroid components . The third analysis is hierarchical, and the TREE procedure is used to display a tree diagram. The results of the analyses follow.
data phys8(type=corr); title 'Eight Physical Measurements on 305 School Girls'; title2 'Harman: Modern Factor Analysis, 3rd Ed, p22'; label height='Height' arm_span='Arm Span' forearm='Length of Forearm' low_leg='Length of Lower Leg' weight='Weight' bit_diam='Bitrochanteric Diameter' girth='Chest Girth' width='Chest Width'; input _name_ $ 1-8 (height arm_span forearm low_leg weight bit_diam girth width)(7.); _type_='corr'; datalines; height 1.0 .846 .805 .859 .473 .398 .301 .382 arm_span.846 1.0 .881 .826 .376 .326 .277 .415 forearm .805 .881 1.0 .801 .380 .319 .237 .345 low_leg .859 .826 .801 1.0 .436 .329 .327 .365 weight .473 .376 .380 .436 1.0 .762 .730 .629 bit_diam.398 .326 .319 .329 .762 1.0 .583 .577 girth .301 .277 .237 .327 .730 .583 1.0 .539 width .382 .415 .345 .365 .629 .577 .539 1.0 ; proc varclus data=phys8; run;
The PROC VARCLUS statement invokes the procedure. By default, PROC VARCLUS clusters on the basis of principal components.
As displayed in Output 78.1.1, the cluster component (by default, the first principal component) explains 58.41% of the total variation in the 8 variables.
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Eight Physical Measurements on 305 School Girls Harman: Modern Factor Analysis, 3rd Ed, p22 Oblique Principal Component Cluster Analysis Cluster Summary for 1 Cluster Cluster Variation Proportion Second Cluster Members Variation Explained Explained Eigenvalue ------------------------------------------------------------------------- 1 8 8 4.67288 0.5841 1.7710 Total variation explained = 4.67288 Proportion = 0.5841 Cluster 1 will be split. Cluster Summary for 2 Clusters Cluster Variation Proportion Second Cluster Members Variation Explained Explained Eigenvalue ------------------------------------------------------------------------- 1 4 4 3.509218 0.8773 0.2361 2 4 4 2.917284 0.7293 0.4764 Total variation explained = 6.426502 Proportion = 0.8033 R-squared with 2 Clusters ----------------- Own Next 1-R**2 Variable Cluster Variable Cluster Closest Ratio Label ----------------------------------------------------------------------------- Cluster 1 height 0.8777 0.2088 0.1545 Height arm_span 0.9002 0.1658 0.1196 Arm Span forearm 0.8661 0.1413 0.1560 Length of Forearm low_leg 0.8652 0.1829 0.1650 Length of Lower Leg ----------------------------------------------------------------------------- Cluster 2 weight 0.8477 0.1974 0.1898 Weight bit_diam 0.7386 0.1341 0.3019 Bitrochanteric Diameter girth 0.6981 0.0929 0.3328 Chest Girth width 0.6329 0.1619 0.4380 Chest Width No cluster meets the criterion for splitting.
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The cluster is split because the second eigenvalue is greater than 1 (the default value of the MAXEIGEN option).
The two resulting cluster components explain 80.33% of the variation in the original variables. The cluster summary table shows that the variables height , arm_span , forearm , and low_leg have been assigned to the first cluster; and that the variables weight , bit_diam , girth , and width have been assigned to the second cluster.
The standardized scoring coefficients in Output 78.1.2 show that each cluster component has similar scores for each of its associated variables. This suggests that the principal cluster component solution should be similar to the centroid cluster component solution, which follows in the next PROC VARCLUS run.
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Oblique Principal Component Cluster Analysis Standardized Scoring Coefficients Cluster 1 2 ------------------------------------------------------------------ height Height 0.266977 0.000000 arm_span Arm Span 0.270377 0.000000 forearm Length of Forearm 0.265194 0.000000 low_leg Length of Lower Leg 0.265057 0.000000 weight Weight 0.000000 0.315597 bit_diam Bitrochanteric Diameter 0.000000 0.294591 girth Chest Girth 0.000000 0.286407 width Chest Width 0.000000 0.272710 Cluster Structure Cluster 1 2 ------------------------------------------------------------------ height Height 0.936881 0.456908 arm_span Arm Span 0.948813 0.407210 forearm Length of Forearm 0.930624 0.375865 low_leg Length of Lower Leg 0.930142 0.427715 weight Weight 0.444281 0.920686 bit_diam Bitrochanteric Diameter 0.366201 0.859404 girth Chest Girth 0.304779 0.835529 width Chest Width 0.402430 0.795572
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The cluster structure table displays high correlations between the variables and their own cluster component. The correlations between the variables and the opposite cluster component are all moderate.
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Oblique Principal Component Cluster Analysis Inter-Cluster Correlations Cluster 1 2 1 1.00000 0.44513 2 0.44513 1.00000
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The intercluster correlation table shows that the cluster components are moderately correlated with =0 . 44513.
In the following statements, the CENTROID option in the PROC VARCLUS statement specifies that cluster centroids be used as the basis for clustering.
proc varclus data=phys8 centroid; run;
The first cluster component, which, in the centroid method, is an unweighted sum of the standardized variables, explains 57.89% of the variation in the data. This value is near the maximum possible variance explained, 58.41%, which is attained by the first principal component (Output 78.1.1).
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Oblique Centroid Component Cluster Analysis Cluster Summary for 1 Cluster Cluster Variation Proportion Cluster Members Variation Explained Explained ---------------------------------------------------------- 1 8 8 4.631 0.5789 Total variation explained = 4.631 Proportion = 0.5789 Cluster Summary for 2 Clusters Cluster Variation Proportion Cluster Members Variation Explained Explained ----------------------------------------------------------- 1 4 4 3.509 0.8773 2 4 4 2.91 0.7275 Total variation explained = 6.419 Proportion = 0.8024 R-squared with 2 Clusters ------------------ Own Next 1-R**2 Variable Cluster Variable Cluster Closest Ratio Label ---------------------------------------------------------------------------------- Cluster 1 height 0.8778 0.2075 0.1543 Height arm_span 0.8994 0.1669 0.1208 Arm Span forearm 0.8663 0.1410 0.1557 Length of Forearm low_leg 0.8658 0.1824 0.1641 Length of Lower Leg ---------------------------------------------------------------------------------- Cluster 2 weight 0.8368 0.1975 0.2033 Weight bit_diam 0.7335 0.1341 0.3078 Bitrochanteric Diameter girth 0.6988 0.0929 0.3321 Chest Girth width 0.6473 0.1618 0.4207 Chest Width
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The centroid clustering algorithm splits the variables into the same two clusters created in the principal component method. Recall that this outcome was suggested by the similar standardized scoring coefficients in the principal cluster component solution.
The default behavior in the centroid method is to split any cluster with less than 75% of the total cluster variance explained by the centroid component. In the next step, the second cluster, with a component that explains only 72.75% of the total variation of the cluster, is split.
In the R-squared table for two clusters, the width variable has a weaker relation to its cluster than any other variable; in the three cluster solution this variable is in a cluster of its own.
Each cluster component (Output 78.1.5) is an unweighted average of the cluster's standardized variables. Thus, the coefficients for each of the cluster's associated variables are identical in the centroid cluster component solution.
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Oblique Centroid Component Cluster Analysis Standardized Scoring Coefficients Cluster 1 2 ------------------------------------------------------------------ height Height 0.266918 0.000000 arm_span Arm Span 0.266918 0.000000 forearm Length of Forearm 0.266918 0.000000 low_leg Length of Lower Leg 0.266918 0.000000 weight Weight 0.000000 0.293105 bit_diam Bitrochanteric Diameter 0.000000 0.293105 girth Chest Girth 0.000000 0.293105 width Chest Width 0.000000 0.293105
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The centroid method stops at the three cluster solution. As displayed in Output 78.1.6 and Output 78.1.7, the three centroid components account for 86.15% of the variability in the eight variables, and all cluster components account for at least 79.44% of the total variation in the corresponding cluster. Additionally, the smallest squared correlation between the variables and their own cluster component is 0.7482.
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Oblique Centroid Component Cluster Analysis Cluster Summary for 3 Clusters Cluster Variation Proportion Cluster Members Variation Explained Explained ----------------------------------------------------------- 1 4 4 3.509 0.8773 2 3 3 2.383333 0.7944 3 1 1 1 1.0000 Total variation explained = 6.892333 Proportion = 0.8615 R-squared with 3 Clusters ------------------ Own Next 1-R**2 Variable Cluster Variable Cluster Closest Ratio Label ---------------------------------------------------------------------------------- Cluster 1 height 0.8778 0.1921 0.1513 Height arm_span 0.8994 0.1722 0.1215 Arm Span forearm 0.8663 0.1225 0.1524 Length of Forearm low_leg 0.8658 0.1668 0.1611 Length of Lower Leg ---------------------------------------------------------------------------------- Cluster 2 weight 0.8685 0.3956 0.2175 Weight bit_diam 0.7691 0.3329 0.3461 Bitrochanteric Diameter girth 0.7482 0.2905 0.3548 Chest Girth ---------------------------------------------------------------------------------- Cluster 3 width 1.0000 0.4259 0.0000 Chest Width
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Oblique Centroid Component Cluster Analysis Total Proportion Minimum Minimum Maximum Number Variation of Variation Proportion R-squared 1-R**2 Ratio of Explained Explained Explained for a for a Clusters by Clusters by Clusters by a Cluster Variable Variable ------------------------------------------------------------------------------------ 1 4.631000 0.5789 0.5789 0.4306 2 6.419000 0.8024 0.7275 0.6473 0.4207 3 6.892333 0.8615 0.7944 0.7482 0.3548
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Note that, if the proportion option were set to a value between 0.5789 (the proportion of variance explained in the 1-cluster solution) and 0.7275 (the minimum proportion of variance explained in the 2-cluster solution), PROC VARCLUS would stop at a two cluster solution, and the centroid solution would find the same clusters as the principal components solution.
In the following statements, the MAXC= option computes all clustering solutions, from one to eight clusters. The SUMMARY option suppresses all output except the final cluster quality table, and the OUTTREE= option saves the results of the analysis to an output data set and forces the clusters to be hierarchical. The TREE procedure is invoked to produce a graphical display of the clusters.
proc varclus data=phys8 maxc=8 summary outtree=tree; run; goptions ftext=swiss; axis2 label=(justify=left); axis1 order=(0.5 to 1.0 by 0.1); proc tree horizontal vaxis=axis2 haxis=axis1 lines=(width=2); height _propor_; id _label_; run;
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Oblique Principal Component Cluster Analysis Total Proportion Minimum Maximum Minimum Maximum Number Variation of Variation Proportion Second R-squared 1-R**2 Ratio of Explained Explained Explained Eigenvalue for a for a Clusters by Clusters by Clusters by a Cluster in a Cluster Variable Variable ---------------------------------------------------------------------------------------- 1 4.672880 0.5841 0.5841 1.770983 0.3810 2 6.426502 0.8033 0.7293 0.476418 0.6329 0.4380 3 6.895347 0.8619 0.7954 0.418369 0.7421 0.3634 4 7.271218 0.9089 0.8773 0.238000 0.8652 0.2548 5 7.509218 0.9387 0.8773 0.236135 0.8652 0.1665 6 7.740000 0.9675 0.9295 0.141000 0.9295 0.2560 7 7.881000 0.9851 0.9405 0.119000 0.9405 0.2093 8 8.000000 1.0000 1.0000 0.000000 1.0000 0.0000
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The principal component method first separates the variables into the same two clusters that were created in the first PROC VARCLUS run. Note that, in creating the third cluster, the principal component method identifies the variable width . This is the same variable that is put into its own cluster in the preceding centroid method example.
The tree diagram in Output 78.1.9 displays the cluster hierarchy. It is clear from the diagram that there are two, or possibly three, clusters present. However, the MAXC=8 option forces PROC VARCLUS to split the clusters until each variable is in its own cluster.
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EAN: 2147483647
Pages: 132